Assume that $R$ is a local ring and that $M$ is a finitely generated $R$-module with a free direct summand, then why its minimal free resolution cannot be periodic?
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If $M$ has a periodic free resolution of period $d\ge 1$, then $M$ is isomorphic to its $d$th syzygy. Now apply Lemma 0.1(ii) in Eisenbud's paper Homological algebra on a complete intersection, with an application to group representations which says that the kernels of the minimal free resolution starting from some rank (depending on $\operatorname{depth} R$) have no free direct summands.